Knot Theory

by Erin Gottschalk, Simon Johnson, Meghan, Elizabeth Nguyen, and Brooke Rogers*.

*Brooke helped the group work through the problem set but was unfortunately unable to attend camp during the blog writing.

What We Did:

Knot theory has many different applications in math including algebra and geometry, and (outside of math) physics. We learned that we can use algebraic techniques to describe knots. When trying to understand knot theory we learned that it is very helpful to work in a group and read the definitions out loud. Us working together was key in understanding knot theory.

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Quantum Mechanics

by Nya Batson, Monique Dacanay, Emily Gao, and Staci Tranquille.

Hello! And welcome to the realm of quantum mechanics!  First off, what in the world is quantum mechanics? Let’s start with a brief introduction.

What is Quantum Mechanics?

Quantum mechanics is one of the most important branches of physics. It focuses on the laws of nature at three different levels: molecular, atomic, and subatomic. Quantum mechanics has a variety of important concepts; the following are some that we learned through our problem set: Planck’s law, the photoelectric effect, and wave-particle duality.  A crucial element of quantum mechanics is understanding that everything has characteristics of both waves and particles. We will touch on this and many other topics later on.

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Network Science

by Kayla Aguilar, Maris James, and Aynsley Szczesniak.

Data is all around us, but it has to be studied in some way, right? How else are we supposed to know what it’s about? That’s what graph theory and network science are for! To organize and connect data mathematicians use networks and graphs as well as scientific computing (like coding).

Network science is an application-based study of graphs. To understand network science, we first have to understand the graphs:

 

Graph Theory

Graphs represent data through nodes, which are the separate points of a graph, and edges, which connect the nodes. There are two types of graphs: directed and undirected graphs. Directed graphs rely on the order of the vertices to be the same, while undirected graphs don’t rely on the order of the nodes.

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Classification of Surfaces

by Camille Clark, Layke Jones, Aekta Kallepalli, Maya Mukerjee, and Caroline Zhou.

 

Euler Characteristics

Euler characteristics

Euler characteristics are defined by the equation V- E + F = 2 where V = number of vertices, E = number of edges or nodes, and F = number of faces. Sometimes though, the equation V – E + F = 2 does not work for all situations because the solution can give various outcomes due to the dimensions and simplicity of the object. If 2 objects are topologically the same, they will have the same Euler characteristics. For all simple polygons, the Euler characteristics equal one. Figures with holes don’t follow these conventions as the holes in these figures add additional faces and edges not proportional to the formulas for simple figures.

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RSA Encryption Cryptography

by Lily Taylor, Zoe Tobien, Tehya Weaver, and Tayloir Wiley.

RSA Encryption Cryptography

What is RSA Encryption Cryptography?

RSA was one of the first public-key* cryptosystems and it is widely used for secure data transmission. It was first created by Ron Rivest, Adi Shamir, and Leona Adleman.

*Public key is used to establish a secret key, and the public key is sent in public. We then use the private key method to encrypt and decrypt large amounts of data, but no one knows the private key.

  • To code: U^s=x X(mod N)=Y
  • To decode Y^t=O O(mod N)=U

In computing, the modulo operation finds the remainder after division of one number by another. Given two positive numbers, a and n, a modulo n (in other words a mod n) is the remainder of the a division of a by n, where a is the dividend and n is the divisor.

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Elliptic Curve Cryptography

by Alana Drumgold*, Lauren Flowers, Emily Huang, Tamarr Moore*, and Ashleigh Sico.

*Alana and Tamarr helped the group work through the problem set but were unfortunately unable to attend camp during the blog writing.

For years, people have been trying to find a way to send secret messages. This may have been easy to do in the ancient times of the Roman Empire, where you could write a message, and then hand-deliver it to your recipient.  This way, you could be certain that nobody else could intercept it. However, this becomes a lot more difficult in today’s online tech-driven world. People no longer hand-deliver letters; rather, we email or text our friends.  So how do we make sure that nobody else can intercept your text message as it travels the internet before finally landing on your friend’s cell-phone? The answer is found in cryptography, a technology that is becoming more and more important in today’s world.  Today, we are going to focus on one particular form of cryptography: elliptic curve cryptography.

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Mathematical Epidemiology

by Jillian Byrnes, La’Ziyah Henry, Delphine Liu, Sophie Ussery, and Elizabeth Whetzel.

What is Mathematical Epidemiology?

What is mathematical epidemiology? Well, mathematical epidemiology is when mathematicians use math to predict outcomes in various statistical problems. These problems include growth in infectious bacteria, change in population, and even the effects of climate change. Why is this used? It is used because it doesn’t need a complete set of data to figure out a solution, as long as you can create an equation and plug in the values.

Who uses it? Mathematicians and scientists use it in fields such as biotechnology, medical science, civil engineering, and as public health professionals.

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Number Systems

by Alysia Davis, Alyssa Drumgold, Pascale Gomez, Delaney Washington, and Auden Wolfe.

 

Intro to Number Systems

As children we grew up counting in the base ten system (1, 2, 3, etc). However, base ten is only one of many numerical systems. Over these past to weeks at Girls Talk Math at UNC, our  task was to explore other number systems that are not as frequently used as the base 10 system, specifically binary and hexadecimal number systems.

 

Binary

The exact definition of binary is related to using a system of numerical notation that has 2 rather than 10 as a base. This means only two single digits are used, 0 and 1. 

Binary is used for data storage. Binary basically makes it easier for computer processors to understand and interpret incoming information/instructions.

Binary was first discussed by Gottfried Leibniz in 1689 but binary numerical systems were not put to use until a binary converter was created hundreds of years later. The binary system was officially implemented just before the beginning of the nineteenth century.

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